Continuation of S-Matrix into Second Riemann Sheet

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Abstract
In the framework of relativistic dispersion theory it is shown that the S-matrix continued into the second Riemann sheet is just the inverse of the one on the first Riemann sheet. This formula is used for deriving a dispersion-like relation between the real and imaginary parts of the scattering phase shift and also the product expansion for the S-matrix. It is noted that these results are generalizations of van Kampen's formulae in the theory of non-relativistic potential scattering to those in the relativistic field theory. The so-called Castillejo-Dalitz-Dyson ambiguity is discussed on the basis of our S-matrix. Relations connecting the sum of the oscillator strengths with scattering lengths are also derived in generalized forms.

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